In 1986 I had a series of conversations with Stephen Wolfram about dynamical systems, but Wolfram didn't tell me where he got his ideas, so I had to do a decade of digging. Here's some of the main points I found documented:
* Dynamics in mathematics is broad and contains the entirety of dynamics from physics. See [*Mathematics Subject Classification 37: Dynamical systems and ergodic theory*](https://zbmath.org/static/msc2020.pdf)
* Dynamics and PDEs are the two mathematical disciplines that are acceptable foundations for physics. When chaos is not present PDEs are fine, but dynamics has evolved to deal with chaos and fractal structures in a way that PDEs can't.
* Dynamical systems are generalizations of iterated functions.
* Dynamics is a collection of mathematical disciplines just as there is no singular chaos theory. Unifying maps and flows is the subject of my research.
* The definition of dynamics @DavidTanzer shared is applicable to any closed system in physics including a QM representation of the entire Universe.